Special bi-invariant linear connections on Lie groups and finite dimensional Poisson structures

Abstract

Let $G$ be a connected Lie group and $\mathfrak{g}$ its Lie algebra. We denote by ${\nabla }^0$ the torsion free bi-invariant linear connection on $G$ given by ${\nabla }_X^{0}Y=\frac{1}{2}[X,Y],$ for any left invariant vector fields $X,Y$. A Poisson structure on $\mathfrak{g}$ is a commutative and associative product on $\mathfrak{g}$ for which ${\mathrm{ad}}_u$ is a derivation, for any $u\in \mathfrak{g}$. A torsion free bi-invariant linear connections on $G$ which have the same curvature as ${\nabla }^0$ is called special. We show that there is a bijection between the space of special connections on $G$ and the space of Poisson structures on $\mathfrak{g}$. We compute the holonomy Lie algebra of a special connection and we show that the Poisson structures associated to special connections which have the same holonomy Lie algebra as ${\nabla }^0$ possess interesting properties. Finally, we study Poisson structures on a Lie algebra and we give a large class of examples which gives, of course, a large class of special connections.

arXiv comment31 pages, This research was conducted within the framework of Action concertée CNRST-CNRS Project SPM04/13